English for Mathematics

Vocabulary

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    accuracy the number of significant figures given in a number; "the atomic clock enabled scientists to measure time with much greater accuracy
    Acute-angled triangle a triangle that has all angles less than 90°
    Addition the process of calculating the total of a group of numbers
    additive inverse one of a pair of numbers whose sum is zero; the additive inverse of -5 is +5
    affine transformation a transformation that is a combination of single transformations such as translation or rotation or reflection on an axis
    Algebraic equation an equation relating to algebra
    Alternate interior angle a pair of angles on the inner side of each of those two lines but on opposite sides of the transversal.
    Approximate value the value that is not completely accurate but close
    Arithmetic the process of making calculations such as adding, multiplying, and dividing by using numbers, or the study of this
    arithmetic progression a progression in which a constant is added to each term in order to obtain the next term; "1-4-7-10-13- is the start of an arithmetic progression
    Associative property the mathematical principle that the order in which three numbers are grouped when being added or multiplied does not matter
    asymmetry, imbalance a lack of symmetry
    Axial relating to or around a real or imaginary straight line going through the centre of an object that is spinning, or a line that divides a symmetrical shape into two equal halves
    axiom a proposition that is not susceptible of proof or disproof; its truth is assumed to be self-evident
    Bayes' postulate the difficulty of applying Bayes' theorem is that the probabilities of the different causes are seldom known, in which case it may be postulated that they are all equal (sometimes known as postulating the equidistribution of ignorance)
    Bayes' theorem a theorem describing how the conditional probability of a set of possible causes for a given observed event can be computed from knowledge of the probability of each cause and the conditional probability of the outcome of each cause
    Bayes’ theorem a way of finding a probability when we know certain other probabilities
    Bernoulli's law, law of large numbers law stating that a large number of items taken at random from a population will (on the average) have the population statistics
    Binary number system a base 2 positional system representing numbers with only the digits 0 and 1
    Binomial distribution the distribution of a random variable denoting the number of successes in a fixed number of independent Bernoulli trials with constant probability
    boundary condition a condition specified for the solution to a set of differential equations
    Cardinal numbers a number like 1, 2, 3 that represents amount, rather than position in a list
    Centimetre a unit of measurement of length in the metric system (= cm)
    Central Limit Theorem the theorem which states that the mean of a large number of independent random variables tends to a normal distribution as long as it satisfies certain conditions.
    Circumference the distance around a circle
    Codomain a superset of the image of the function. It is sometimes known as the range of the function.
    Coefficient a value, in mathematics, that appears in front of and multiplies another value
    Commutative property a calculation that gives the same result whatever order the values are in
    completeness an attribute of a logical system that is so constituted that a contradiction arises if any proposition is introduced that cannot be derived from the axioms of the system
    Composite number a positive number (= larger than zero) that can be divided by positive numbers other than 1 and itself
    Composition the act of combining 2 mathematical objects in some way
    Concavity the quality of curving in, or an object or surface that curves in
    Concyclic having the property of being on the circumference of a circle.
    Conditional probability the probability of an event given the (non-)occurrence of other events.
    Cone a solid shape with a round base that narrows to a point at the top
    Confidence interval an interval given as the estimate for a parameter, based on the theoretic value of the parameter given known information, while taking into account of the probability we require the actual parameter to be within the given interval.
    Congruent triangles two or more triangles that have the same size and shape
    conic section, conic a curve generated by the intersection of a plane and a circular cone
    consistency an attribute of a logical system that is so constituted that none of the propositions deducible from the axioms contradict one another
    Constant a particular number or amount that never changes
    Continuous data data that can take any value (within a range)
    contradiction, contradiction in terms a statement that is necessarily false; "the statement `he is brave and he is not brave' is a contradiction
    corollary an inference that follows directly from the proof of another proposition
    Corresponding angle an angle in matching corners
    Cosecant a function (= a mathematical relation) of an angle that is the reciprocal of sine
    Cosine (in a triangle that has one angle of 90°) the ratio of the length of the side next to an angle less than 90°, divided by the length of the hypotenuse (= the side opposite the 90° angle)
    Cotangent a function (= a mathematical relation) of an angle that is the reciprocal (= number) of tangent
    Cryptography the practice of creating and understanding codes that keep information secret
    Cube a solid object with six square sides of equal size
    Cubic centimetre a measurement equal to the volume of a cube (= solid object with six equal sides) measuring one centimetre on each edge (= cm3)
    Cubic metre a measurement equal to the volume of a cube (= solid object with six equal sides) measuring one metre on each edge (= m3)
    Cuboid a solid object with six rectangular sides
    Cyclic quadrilateral a quadrilateral for which a circle can be circumscribed so that it touches each polygon vertex
    Cylinder a solid that has long straight sides and circular ends of equal size, or a hollow object shaped like this and often used as a container
    decile any of nine points that divided a distribution of ranked scores into equal intervals where each interval contains one-tenth of the scores
    Decimal a number expressed in a particular positional numbering system of base 10
    Decomposition the process of splitting a mathematical object into two or more objects of the same type
    degree of freedom an unrestricted variable in a frequency distribution
    Denary number system a positional number system of base 10 or the concept of representing numbers using base 10 (independent of symbolic representations).
    Denominator in a fraction (= part of a whole number), the number written below the line, showing how many parts the whole contains
    Dependent variable a number or amount whose value depends on the value of another element in the same mathematical expression (= group of symbols representing an amount or idea)
    Determinant a scalar quantity associated with a square matrix, or the transformation represented by the square matrix is a certain coordinate system.
    diagonal a straight line connecting any two vertices of a polygon that are not adjacent
    Differentiation the process of calculating the derivative of a function
    Direct variation a relationship between two variables x and y that can be written as y = kx, k ≠ 0.
    Discrete data data that can only take certain values
    Discriminant the sum of the squares of differences between all roots of a polynomial
    Dispersion the variability of a quantity. In this sense, a data set can be seen as multiple instances of a variable.
    distribution, statistical distribution an arrangement of values of a variable showing their observed or theoretical frequency of occurrence
    Distributive property the property of an operation on another where (for each element) the order of the operations are commutative
    Division the calculation of how many times one number goes into another
    Domain The set of values as arguments for a function for which the values are defined
    duality the interchangeability of the roles of points and planes in the theorems of projective geometry
    eccentricity a ratio describing the shape of a conic section; the ratio of the distance between the foci to the length of the major axis; "a circle is an ellipse with zero eccentricity
    Edge a line segment between two vertices in a graph (graph theory) or geometric shape.
    Equation a mathematical statement that two amounts, or two symbols or groups of symbols representing an amount, are equal
    Equilateral triangle a triangle that has all sides the same length
    Euler’s formula an identity relating the constant e to the trigonometric functions eix = cos x + i sin x
    Even numbers forming a whole number that can be divided exactly by two
    Expectation also known as an expected value. The average (mean) of the values of a random variable weighted by the probability
    explanans statements that explain the explicandum; the explanatory premises
    explicandum, explanandum a statement of something (a fact or thing or expression) to be explained
    Exponential function a function containing an exponent (= a number or sign that shows how many times another number is to be multiplied by itself)
    extrapolation calculation of the value of a function outside the range of known values
    Face The plane figure between (at least 3) adjacent edges of a polyhedron where all the edges lie on the extended plane (from the plane figure) and all other edges lie on only one side of the plane.
    Factor any whole number that is produced when you divide a larger number by another whole number
    Factor theorem a theorem that relates the roots of a polynomial to its factors. class="d-title" Namely, that (x - a) is a factor of the polynomial p(x) as long as a is a root of p(x). (i.e., p(a)=0) The converse is also true though trivial. The Factor Theorem can be considered as a special case of the remainder theorem.
    factorization, factorisation, factoring the resolution of an integer or polynomial into factors such that when multiplied together they give the integer or polynomial
    field a set of elements such that addition and multiplication are commutative and associative and multiplication is distributive over addition and there are two elements 0 and 1; "the set of all rational numbers is a field
    fractal a geometric pattern that is repeated at every scale and so cannot be represented by classical geometry
    Fraction a number that results from dividing one whole number (= a number with no part of a number after it) by another
    Function a quantity whose value depends on another value and changes with that value
    geodesic, geodesic line the shortest line between two points on a mathematically defined surface (as a straight line on a plane or an arc of a great circle on a sphere)
    Geometric consisting of shapes such as squares, triangles, or rectangles
    geometric progression a progression in which each term is multiplied by a constant in order to obtain the next term; "1-4-16-64-256- is the start of a geometric progression
    Global extrema the largest or smallest values of the entire function
    Gram a basic unit of weight in the metric system (= g)
    Greatest common divisor the greatest value in the set of common factor between two numbers (= H.C.F.)
    harmonic progression a progression of terms whose reciprocals form an arithmetic progression
    Heron’s formula an equation relating the area of a triangle to its sides: with a, b, and c are the lengths of the sides of the triangle and s is half of the perimeter (sum of the lengths of sides)
    Hexadecimal number system a base -16 number system. The English letters A, B, C, D, E and F are usually used for the digits 10, 11, 12, 13, 14 and 15 in such a system.
    Hexagon a flat shape with six straight sides
    Highest Common Factor (H.C.F.) the greatest value in the set of common factor between two numbers
    Histogram a bar chart
    Identities a mathematical statement with the symbol ≡ that express the equality of the two expressions on either side of it under all circumstances, i.e. no matter what value the variables take. It is a stronger statement than an equation.
    Inclination the angle at which something slopes
    inclination, angle of inclination the angle formed by the x-axis and a given line (measured counterclockwise from the positive half of the x-axis)
    Independent variable a number or amount whose value does not depend on the value of another element in the same mathematical expression (= group of symbols representing an amount or idea)
    independent variable, experimental variable a variable whose values are independent of changes in the values of other variables
    Inequalities any mathematical sentences that states the relationship between 2 values or expressions by asserting that: 1) They are not the same 2) That one is greater than the other. / That one is less than the other. 3) That one is not greater than the other. / That one is not less than the other. Through the use of symbols ≠, >, ≥, < or≤.
    Integration the inverse process of differentiation, as related by the fundamental theorem of calculus.
    Interior angle an angle inside a shape
    interpolation calculation of the value of a function between the values already known
    Intersection a point or set of points where two lines, planes, etc., cross
    Inverse variation for two quantities with inverse variation, as one quantity increases, the other quantity decreases, written as xy = k, x ≠ 0, y ≠ 0.
    Irrational number a number that cannot be expressed as the ratio of two whole numbers
    Isosceles right-angled triangle a triangle that has one angle of 90°and two sides of equal length
    Isosceles triangle a triangle with two sides of equal length
    Joint variation a situation where one variable depends on two (or more) other variables, and varies directly as each of them when the others are held constant. We say z varies jointly as x and y if z = kxy, k ≠ 0.
    Kilogram a basic unit of weight in the metric system (= kg)
    Kilometre a unit of measurement of length in the metric system (= km)
    Least Common Multiple (L.C.M.) The smallest of the set of numbers which are multiples of all the numbers within a given set of positive numbers
    Limit a mathematical object perceived to represent the "last" object in a sequence of the same type of mathematical object in the sense that there are truncated sequences for which all objects are within any specified neighbourhood of the limit.
    Linear equation an equation (= a mathematical statement) with a result that, when it is put on a graph, forms a straight line
    Litre a unit for measuring liquids or gases in the metric system (= L)
    Local extrema the smallest or largest outputs of a small part of the function
    Logarithms a number which shows how many times a particular number, called the base, has to be multiplied by itself to produce another number
    lower bound a number equal to or less than any other number in a given set
    Marked price the price of an article for sale as indicated by an attached label, etc.
    Mathematical Induction a technique for proving results or establishing statements for natural numbers.
    mathematical process, mathematical operation, operation calculation by mathematical methods; "the problems at the end of the chapter demonstrated the mathematical processes involved in the derivation"; "they were learning the basic operations of arithmetic
    mathematical space, topological space any set of points that satisfy a set of postulates of some kind; "assume that the topological space is finite dimensional
    Matrix a group of numbers or other symbols arranged in a rectangle that can be used together as a single unit to solve particular mathematical problems
    Mean the result you get by adding two or more amounts together and dividing the total by the number of amounts (= average)
    Median the middle one in a set of values arranged in order of size
    Metre a unit of measurement of length in the metric system (= m)
    Millilitre a unit for measuring liquids or gases in the metric system (= mL)
    Millimetre a unit of measurement of length in the metric system (= mm)
    Mode / Modal class the number or value that appears the most often in a particular set of numbers or values
    Multiple a number that results from multiplying one number by another when at least one of them is a whole number (= a number with no part of a number after it)
    Multiplication the process of adding a number to itself a particular number of times, or a calculation in which this is done
    multiplicative inverse, reciprocal one of a pair of numbers whose product is 1: the reciprocal of 2/3 is 3/2; the multiplicative inverse of 7 is 1/7
    negation a proposition that is true if and only if another proposition is false
    non sequitur a conclusion that does not follow from the premises
    Normal distribution the limit of a binomial distribution, as the number of trials tend to infinity
    Numerator the number above the line in a fraction
    Obtuse-angled triangle a triangle in which one of the interior angles measures more than 90°
    Odd numbers numbers that are not able to be divided exactly by two
    operator a symbol that represents a function from functions to functions; "the integral operator
    Ordinal numbers a number such as 1st, 2nd, 3rd, 4th, that shows the position of something in a list of things
    osculation a contact of two curves (or two surfaces) at which they have a common tangent
    Parabola a type of curve such as that made by an object that is thrown up in the air and falls to the ground in a different place
    paradox a self-contradiction; "`I always lie' is a paradox because if it is true it must be false
    Parallelogram a flat shape that has four sides. The two sets of opposite sides are parallel and of equal length to each other
    parity a relation between a pair of integers: if both integers are odd or both are even they have the same parity; if one is odd and the other is even they have different parity
    Pentagon a shape with five sides and five angles
    percentile, centile any of the 99 numbered points that divide an ordered set of scores into 100 parts each of which contains one-hundredth of the total
    Perimeter the perimeter is the outer edge of a flat shape or area
    Periodicity the tendency of an event or series of events to happen repeatedly in a fixed pattern
    Permutation any of the various ways in which a set of things can be ordered
    Perpendicular at an angle of 90° to another line or surface
    Pictogram a type of graph that uses pictures or symbols to show or compare data
    Plane figures a geometric figure that has no thickness
    plane, sheet an unbounded two-dimensional shape; "we will refer to the plane of the graph as the X-Y plane"; "any line joining two points on a plane lies wholly on that plane
    Poisson distribution the discrete probability distribution of the number of independently singly events over a fixed "length" (of space or time or similar continua) given a constant rate.
    Polyhedron a solid shape with four or more flat surfaces
    Polynomial a number or variable (= mathematical symbol), or the result of adding or subtracting two or more numbers or variables
    population, universe the entire aggregation of items from which samples can be drawn; "it is an estimate of the mean of the population
    postulate, posit a proposition that is accepted as true in order to provide a basis for logical reasoning
    postulation, predication a declaration of something self-evident; something that can be assumed as the basis for argument
    Power the number of times that a number is to be multiplied by itself
    predicate what is predicated of the subject of a proposition; the second term in a proposition is predicated of the first term by means of the copula; "`Socrates is a man' predicates manhood of Socrates
    Prime number a number that cannot be divided by any other number except itself and the number 1
    Prism a transparent glass or plastic object that separates white light that passes through it into different colours
    Probability a number that represents how likely it is that a particular thing will happen
    proposition a statement that affirms or denies something and is either true or false
    Protractor a device used for measuring and drawing angles. It is usually in the form of half a circle made from transparent plastic with degrees printed on it
    Pyramid a solid shape with a flat, square base and four flat, triangular sides which slope inward and meet to form a point at the top
    Pythagoras’ theorem (= a statement that in a right-angled triangle (= a triangle with a 90° angle) the square of the length of the side opposite the 90° angle is equal to the squares of each of the other two sides added together)
    Quadratic equation an equation that includes an unknown value that is multiplied by itself only once, and does not include an unknown value multiplied by itself more than once
    Quadratic surds an expression containing square roots, such that the number under the square root is a rational number and is not a perfect square
    Quadrilateral a flat shape with four straight sides
    quantifier, logical quantifier a word (such as `some' or `all' or `no') that binds the variables in a logical proposition
    quartile any of three points that divide an ordered distribution into four parts each containing one quarter of the scores
    Quotient rule a method of finding the derivative of a function that is the ratio of two differentiable functions
    Radioactive decay the process by which an unstable atomic nucleus loses energy by radiation
    Radius (the length of) a straight line from the centre of a circle to its edge
    Rational number a number that can be expressed as the ratio of two whole numbers
    rationalization, rationalisation the simplification of an expression or equation by eliminating radicals without changing the value of the expression or the roots of the equation
    ray a straight line extending from a point
    recursion an expression such that each term is generated by repeating a particular mathematical operation
    recursive definition a definition of a function from which values of the function can be calculated in a finite number of steps
    reflection a transformation in which the direction of one axis is reversed
    reflexivity a relation such that it holds between an element and itself
    Remainder the amount that is left when one number cannot be exactly divided by another
    Remainder theorem a theorem which states that f(k) is the number to take away from the polynomial f(x) so that the quotient of dividing the resulting polynomial by (x - k) is another polynomial.
    Rhombus a flat shape that has four sides that are all of equal length
    Right-angled triangle a triangle that has one angle of 90°
    Root a solution of some equations
    rotation a transformation in which the coordinate axes are rotated by a fixed angle about the origin
    rounding, rounding error a miscalculation that results from rounding off numbers to a convenient number of decimals; "taxes are rounded off to the nearest dollar but the rounding error is surprisingly small
    rule, formula a standard procedure for solving a class of mathematical problems; "he determined the upper bound with Descartes' rule of signs"; "he gave us a general formula for attacking polynomials
    sampling the selection of a suitable sample for study
    Scalar something that has size but no direction, such as a quantity, distance, speed, or temperature
    Scalene triangle a triangle with three sides all of different lengths
    Secant in a triangle that has one angle of 90°, the ratio of the hypotenuse (= the length of the side opposite the 90° angle) to the length of the side next to an angle less than 90°
    Second derivative the derivative of the derivative of a function f(x), denoted f''(x).
    section, plane section the area created by a plane cutting through a solid
    series the sum of a finite or infinite sequence of expressions
    set an abstract collection of numbers or symbols; "the set of prime numbers is infinite
    Simultaneous equation a set of equations where the solution set contains the same value for the same variables
    Sine (in a triangle that has one angle of 90°) the ratio of the length of the side opposite an angle less than 90° divided by the length of the hypotenuse (= the side opposite the 90° angle)
    Slope a measure of how steep an angle a line has
    Sphere a solid shape like a round ball
    square, foursquare a plane rectangle with four equal sides and four right angles; a four-sided regular polygon; "you can compute the area of a square if you know the length of its sides
    Standard deviation a number that shows the amount by which members of a group are different from the mean (= average) value for the group
    Stem-and-leaf diagram a diagram that quickly summarizes data while maintaining the individual data points
    subgroup a subset (that is not empty) of a mathematical group
    subject the first term of a proposition
    Subtraction the process of taking one number or amount away from another number or amount
    Summation the process of finding the sum of a sequence of quantities.
    superposition the placement of one object ideally in the position of another one in order to show that the two coincide
    symmetry, symmetricalness, correspondence, balance an attribute of a shape or relation; exact correspondence of form on opposite sides of a dividing line or plane
    Tangent (in a triangle that has one angle of 90°) the ratio of the length of the side opposite an angle less than 90° divided by the length of the shorter of the two sides that are next to the angle
    tautology a statement that is necessarily true; "the statement `he is brave or he is not brave' is a tautology
    Tetrahedron a solid shape whose four surfaces are triangles
    Theorem a formal statement that can be shown to be true by logic
    transformation a function that changes the position or direction of the axes of a coordinate system
    transitivity a relation between three elements such that if it holds between the first and second and it also holds between the second and third it must necessarily hold between the first and third
    translation a transformation in which the origin of the coordinate system is moved to another position but the direction of each axis remains the same
    transposition the transfer of a quantity form one side of an equation to the other along with a change of sign
    Trapezium a flat shape with four sides, where two of the sides are parallel
    Trigonometric equation a function relating to trigonometry (= a type of mathematics that deals with the relationship between the angles and sides of triangles)
    Trigonometry a type of mathematics that deals with the relationship between the angles and sides of triangles
    truncation error a miscalculation that results from cutting off a numerical calculation before it is finished
    upper bound a number equal to or greater than any other number in a given set
    Variance a measure of dispersion, calculated by finding the mean of the square of numbers and the square of the mean of numbers
    Vector a representation of something that has both direction and size, usually an arrow whose direction represents direction and length represents size
    Vector product an vector operation in three-dimensional Euclidean space.
    Venn Diagram a diagram style that shows the logical relation between sets
    Vertex the point where two lines meet to form an angle, or the point that is opposite the base of a shape
    weight, weighting a coefficient assigned to elements of a frequency distribution in order to represent their relative importance

    Short Quiz

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    Inference in Principal Component Analysis (PCA) is classically based on the asymptotic ______ of the top eigenvalues of the sample covariance matrix, which are consistent estimators of the top ______ eigenvalues under low-dimensional asymptotics—when the sample size grows while the dimension is fixed (Anderson, 1963, 2003).

     

    (Excerpt from: Dobriban, Edgar. (2017). Sharp Detection in PCA under Correlations: All Eigenvalues Matter. The Annals of Statistics, 45(4), 1810-1833.) 

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